# casinf, casin, casinl

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< c‎ | numeric‎ | complex

Defined in header ``<complex.h>`` |  |
float complex casinf( float complex z ); | (1) | (since C99)
double complex casin( double complex z ); | (2) | (since C99)
long double complex casinl( long double complex z ); | (3) | (since C99)
Defined in header ``<tgmath.h>`` |  |
#define asin( z ) | (4) | (since C99)

1-3) Computes the complex arc sine of `z` with branch cuts outside the interval [−1,+1] along the real axis.
4) Type-generic macro: If `z` has type long double complex, `casinl` is called. if `z` has type double complex, `casin` is called, if `z` has type float complex, `casinf` is called. If `z` is real or integer, then the macro invokes the corresponding real function (asinf, asin, asinl). If `z` is imaginary, then the macro invokes the corresponding real version of the function asinh, implementing the formula \(\small \arcsin({\rm i}y) = {\rm i}{\rm arsinh}(y)\)arcsin(iy) = i arsinh(y), and the return type of the macro is imaginary.

### Parameters

z | - | complex argument

### Return value

If no errors occur, complex arc sine of `z` is returned, in the range of a strip unbounded along the imaginary axis and in the interval [−π/2; +π/2] along the real axis.

Errors and special cases are handled as if the operation is implemented by -I * casinh(I*z)

### Notes

Inverse sine (or arc sine) is a multivalued function and requires a branch cut on the complex plane. The branch cut is conventionally placed at the line segments (-∞,-1) and (1,∞) of the real axis.

The mathematical definition of the principal value of arc sine is \(\small \arcsin z = -{\rm i}\ln({\rm i}z+\sqrt{1-z^2})\)arcsin z = -iln(iz + √1-z2
)

For any z, \(\small{ \arcsin(z) = \arccos(-z) - \frac{\pi}{2} }\)arcsin(z) = acos(-z) -

π
2

### Example

Run this code

    #include <stdio.h>
    #include <math.h>
    #include <complex.h>

    int main(void)
    {
        double complex z = casin(-2);
        printf("casin(-2+0i) = %f%+fi\n", creal(z), cimag(z));

        double complex z2 = casin(conj(-2)); // or CMPLX(-2, -0.0)
        printf("casin(-2-0i) (the other side of the cut) = %f%+fi\n", creal(z2), cimag(z2));

        // for any z, asin(z) = acos(-z) - pi/2
        double pi = acos(-1);
        double complex z3 = csin(cacos(conj(-2))-pi/2);
        printf("csin(cacos(-2-0i)-pi/2) = %f%+fi\n", creal(z3), cimag(z3));
    }

Output:

    casin(-2+0i) = -1.570796+1.316958i
    casin(-2-0i) (the other side of the cut) = -1.570796-1.316958i
    csin(cacos(-2-0i)-pi/2) = 2.000000+0.000000i

### References

- C11 standard (ISO/IEC 9899:2011):

  - 7.3.5.2 The casin functions (p: 190)

  - 7.25 Type-generic math <tgmath.h> (p: 373-375)

  - G.7 Type-generic math <tgmath.h> (p: 545)

- C99 standard (ISO/IEC 9899:1999):

  - 7.3.5.2 The casin functions (p: 172)

  - 7.22 Type-generic math <tgmath.h> (p: 335-337)

  - G.7 Type-generic math <tgmath.h> (p: 480)

### See also

cacoscacosfcacosl (C99)(C99)(C99) | computes the complex arc cosine (function)
catancatanfcatanl (C99)(C99)(C99) | computes the complex arc tangent (function)
csincsinfcsinl (C99)(C99)(C99) | computes the complex sine (function)
asinasinfasinl (C99)(C99) | computes arc sine (\({\small\arcsin{x} }\)arcsin(x)) (function)
C++ documentation for asin

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